In this paper, multiple cores sandwich composites undergoing impact loads are optimized in order to improve their resistance to the impact-induced delamination. This peculiar type of composites is characterized by one internal face splitting the core in two parts. Owing to their architecture with an intermediate and two external faces, their additional tailoring capability offers potential advantages in terms of energy absorption capability and damage tolerance behavior over conventional sandwich composites. Obviously, an accurate assessment of the interfacial stress fields, of their damage accumulation mechanisms and of their post-failure behavior are fundamental to fully exploit their potential advantages. Despite it is evident that structural models able to accurately describe the local behavior are needed to accomplish this task, the analysis is commonly still carried out using simplified sandwich models which postulate the overall variation of displacements and stresses across the thickness, because more detailed models could make the computational effort prohibitively large. No attempt is here made to review the ample literature about the sandwich composite models, since a plenty of comprehensive bibliographical review papers and monographs are available in the specialized literature. Likewise, no attempt is made for reviewing the methods used to model the damage. It is just remarked that the models to date available range from detailed models which discretize the real structure of the core, to FEM models by brick elements, to discrete-layer models and to sublaminate models. In these paper, two different models are used, to achieve a compromise between accuracy and limitation of costs. The time history of the contact force is computed by a C° eight-node plate element based on a 3D zig-zag model, in order to achieve the best accuracy using a plate model with the customary five functional d.o.f. This model is also used in the optimization process, since it is mathematically easily treatable and accurately describes the strain energy. In addition, it enables a comparison with the classical plate models, since they can be particularized from it. The counterpart plate element of this zig-zag model, which is obtained from a standard C° plate element through a strain energy updating (which successfully described the impact induced damage as shown by the comparison with the damage detected by c-scanning in a previous paper), is used for computing the contact force time history, to reach a good compromise between accuracy and computational costs. A mixed brick element with the three displacements and the three interlaminar stresses as nodal d.o.f. is used to compute the damage at each time step. The onset of damage is predicted in terms of matrix and fibers failure, cracks, delamination, rippling, wrinkling and face damping using different stress-based criteria. In this paper the effects of the accumulated damage are accounted for through the ply-discount theory, i.e. using reduced elastic properties for the layers and the cores that failed, although it is known that some cases exist for which this material degradation model could be unable to describe the real loss of load carrying capacity. The optimization technique recently proposed by the authors is used in this paper for optimizing the energy absorption properties of multi-core sandwiches undergoing impact loads. The effect of this technique is to act as an energy absorption tuning, since it minimizes or maximizes the amount of energy absorbed by specific modes through a suited in-plane variation of the plate stiffness properties (e.g., bending, in-plane and out-of-plane shears and membrane energies). The appropriate in-plane variable distributions of stiffness properties, making certain strain energy contributions of interest extremal, are found solving the Euler-Lagrange equations resulting from assumption of the laminate stiffness properties as the master field and setting to zero the first variation of wanted and unwanted strain energy contributions (e.g., bending, in-plane and out-of-plane shears and membrane energies). Our purpose is to minimize the energy absorbed through unwanted modes (i.e., involving interlaminar strengths) and maximize that absorbed through desired modes (i.e., involving membrane strengths). The final result is a ply with variable stiffness coefficient over its plane which is able to consistently reduce the through-the-thickness interlaminar stress concentrations, with beneficial effects on the delamination strength. All the solutions proposed can be obtained either varying the orientation of the reinforcement fibers, the fiber volume rate or the constituent materials by currently available manufacturing processes. The coefficients of the involved stiffness terms are computed enforcing conditions which range from the thermodynamic constraints, to imposition of the mean stiffness, to the choice of a convex or a concave shape (in order to minimize or maximize the energy contributions of interest). Two solutions of technical interest will be proposed, which both are based on a parabolic distribution of stiffness coefficients. The former reduces the bending of a lamina with moderately increasing the shear stresses, the second one reduces these stresses with a low increment in the bending contribution. The effects of the incorporation of these layers (with the same mean properties of the layers they replace) is shown hereafter.
Phononic Band Gaps of Elastic Periodic Structures: A Homogenization Theory Study